Quantum Theory: Operator Exponentiation

Homework Statement

Let [itex]\left|x\right\rangle[/itex] and [itex]\left|p\right\rangle[/itex] denote position and momentum eigenstates, respectively. Show that [itex]U^n\left|x\right\rangle[/itex] is an eigenstate for [itex]x[/itex] and compute the eigenvalue, for [itex]U = e^{ip}[/itex]. Show that [itex]V^n\left|p\right\rangle[/itex] is an eigenstate for [itex]p[/itex] and compute the eigenvalue, for [itex]V = e^{ix}[/itex].

The Attempt at a Solution

I know that [itex](e^{ip})^{n} = e^{inp}[/itex], since p obviously commutes with itself; I also know that momentum is defined as the generator of translations, which leads to a translation operator [itex]T(x') = e^{(\frac{ipx'}{\hbar})}[/itex] with the property that [itex]T(x') \left| x \right\rangle = \left|x+x'\right\rangle[/itex], which is also an eigenstate of [itex]x[/itex]. Is the solution as simple as identifying [itex] U^n [/itex] with the translation operator, with [itex]n[/itex] in units of length/action?

Homework Statement

Let [itex]\left|x\right\rangle[/itex] and [itex]\left|p\right\rangle[/itex] denote position and momentum eigenstates, respectively. Show that [itex]U^n\left|x\right\rangle[/itex] is an eigenstate for [itex]x[/itex] and compute the eigenvalue, for [itex]U = e^{ip}[/itex]. Show that [itex]V^n\left|p\right\rangle[/itex] is an eigenstate for [itex]p[/itex] and compute the eigenvalue, for [itex]V = e^{ix}[/itex].

The Attempt at a Solution

I know that [itex](e^{ip})^{n} = e^{inp}[/itex], since p obviously commutes with itself; I also know that momentum is defined as the generator of translations, which leads to a translation operator [itex]T(x') = e^{(\frac{ipx'}{\hbar})}[/itex] with the property that [itex]T(x') \left| x \right\rangle = \left|x+x'\right\rangle[/itex], which is also an eigenstate of [itex]x[/itex]. Is the solution as simple as identifying [itex] U^n [/itex] with the translation operator, with [itex]n[/itex] in units of length/action?

Hint: Consider the action of the commutator [itex][U^n, p][/itex] on the ket [itex]|x\rangle[/itex]